12 Answers
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I think it's important to look at (one of) the Joker's origin stories. From The Killing Joke:

You see it doesn't matter if you catch me and send me back to the asylum... [...] I've proved my point. I've demonstrated there's no difference between me and everyone else! All it takes is one bad day to reduce the sanest man alive to lunacy. That's how far the world is from where I am. Just one bad day. You had a bad day once, am I right? I know I am. I can tell. You had a bad day and everything changed. Why else would you dress up as a flying rat? You had a bad day, and it drove you as crazy as everybody else... [...] Something like that happened to me, you know. [...] My point is, I went crazy. When I saw what a black, awful joke the world was, I went crazy as a coot!

You can see this philosophy behind so much of what the Joker does throughout the movie, notice how many of his crimes involve getting relatively innocent people to commit atrocities. He does whatever he can to give them that one bad day that will push them over the edge.

While Batman's sufferings have definitely left him alienated from the regular populace (stranger), they've also made him the man he is (stronger). The Joker's goal is just to corrupt people, to leave them as twisted and messed up as he is. This is particularly evident with Harvey Dent to whom the Joker gives personal attention. You can see his glee in corrupting Gotham's "White Knight".

Basically, while the Joker has no problem with killing people, he loves it when he can make them as strange as he is. Most of his plans are set up so that one way or another, he'll get an outcome that makes him happy.

This means that every experience in life makes you a stronger, more rounded person. Joker's phrase is obviously a word play on this, as other answered have already identified, but to give some more reasons on why Joker says it, you have to understand his character a little better. To use his excellent Wikipedia article:

The Joker's main characteristic is his apparent insanity, although he
is not described as fitting any particular psychological disorder. He
displays a lack of conscience and empathy, and no concern over right
and wrong demonstrating extreme psychopathy. In A Serious House on
Serious Earth, Joker is described as only being capable of processing
sensory information from the outside world by simply adapting to it,
causing him to create a new personality every day depending on what
would benefit him most, explaining why he is sometimes a mischievous
clown and at others a psychopathic killer.

...Much like the Joker wears no mask and simply is the Joker, he believes
Batman is Batman with or without the costume, and repeatedly ignores
attempts to learn his true identity as he has no interest in what lies
behind Batman's mask. Similarly, when given the opportunity
to kill Batman, Joker opts not to, believing that without their game,
winning is pointless. The Joker has no desire for typical
criminal goals like money or power, and his acts are designed only to
continue his game with Batman.

Joker is a bizarre, twisted individual who adores having "fun" with Batman, whilst simultaneously causing mass chaos and havoc on Gotham City. In the Christopher Nolan Trilogy in particular, the strange nature of Joker was emphasised. Think of some of the scenes he had like dressing up as a nurse at the hospital, or burning the mountain of money. His strange attitude is one of the things that make him so terrifyingly brilliant as a character - he wants nothing. He can't be reasoned or bartered with. He simply wants chaos.

The word play given by Joker is just one of many examples in the movie of his unusual and weird character.

“It's quite possible we may actually be looking at some kind of super-sanity here. A brilliant new modification of human perception, more suited to urban life at the end of the twentieth century...He creates himself each day. He sees himself as the lord of misrule and the world as a theatre of the absurd.” ― Grant Morrison, Batman: Arkham Asylum
– Peter HanleyMar 14 '14 at 15:42

As noted in other answers, it's wordplay and emphasizes how strange the Joker himself is. But the literal meaning of the expression is also important; to me, it seems quite reasonable to assume that the Joker really does believe that traumatic experiences ("whatever doesn't kill you") can push people to extremes and cause them to abandon social norms ("makes you stranger"). In particular, he thinks that even very stalwart-seeming people, those who attempt to uphold that which they consider "good," "virtuous," or "just," will eventually give up their socially normative ethics if they are pushed hard enough.

In other words, the point isn't so much that the Joker himself is "screwed up" by his own experiences; the point is that he believes that he can "screw up" everyone else around him using psychological warfare. His behavior, his individual crimes, and his overarching strategies are all motivated by his efforts to "break" people, both individually and en masse:

A substantial portion of the film is dedicated to showing his efforts to force Batman into killing someone, which would mean that Batman would break his "one rule."

His threat to blow up the hospital is a ploy to manipulate random Gotham citizens to commit murder.

Similarly, the "game" on the boats is an attempt to show that when given an incentive, groups of people will agree to blow up other groups of people.

While being interrogated by Batman, he's reveling in how far he's pushed this "good" man emotionally--"look at you go!"

And, of course, the "ace" up his sleeve is Harvey Dent, Gotham's "White Knight," a paragon of (legal) justice, whom he has successfully turned into a murderer.

In my opinion what he means and it's way deeper than you though is that what you live in life and specially the fact of being the Batman makes you stranger to yourself and your friend because it's transforming you.
– KiwyMar 13 '14 at 12:00

It also makes you stranger in a much simpler way. If you survive a fall that should have killed you, most people would consider that strange.
– David StarkeyMar 13 '14 at 14:46

1

@Kiwy I'm not entirely sure what you're trying to say ("deeper than you"?), but why focus on Batman in particular? As I pointed out, the Joker seems to believe everyone can be made "strange."
– Kyle StrandMar 13 '14 at 16:27

It was a simple wordplay. He took a very famous and often used phrase (that you correctly identified in your question) and made it, well, stranger. This is just the Joker's kind of humour, especially when seeing that he has a thing for taking the normal and driving it ad absurdum. So I'm not sure he intended anything else than being funny and, uh, strange.

For the joker, who believes he is immortal, or at least behaves as if he is, life would be boring without chaos, the strange, or unexpected is the root of chaos. The more chaos, the more victorious or strong he feels. Nothing seems to kill him, so whatever entertaining means that others use in attempts to kill him only satisfy him in watching the chaos. Stranger, to the Joker, IS stronger.

I think he may have been alluding to his scars. He takes his mask off right when he says "..stranger." and then smiles and gets up. Maybe he was saying that though he wasn't killed when he was deformed, it did make him "stranger" in a sense.

I love the deep answers, about Jokers background. But I don't think it's really that deep of a meaning. As any other character, in any other movie could say the same thing, and would make just as much sense.

If someone was injured/traumatized, physically and/or mentally, there's a good chance they'll be "stranger" after the incident, assuming they lived.

For example:

If someone had fallen off a building and broke both legs. They may now have a "strange" gait.

A burn victim, may look "strange", as their skin is now stretched and grafted.

Someone who's been abused, may later exhibit "strange" behavior. Such as PTSD, ticks, emulation of their abuser, etc.

This line is obviously a play on Nietzsche's, but it also implies that he's just someone normal but who has lived some very unusual, traumatic events (this is backed by the Alan Moore's backstory of th Joker).

He's implying that anyone can be like him, he's just anyone. Just like Batman, and is thus making a direct connection between them, implying that they're just the same.

This idea was also used in Batman Asylum game with the mythical image of Cain and Abel and the end of the game when Batman is carrying the dead body of his "brother" Joker.

Traumas no matter they are are physical, mental, emotional or of any other kind changes the persons physical or emotional being or both, thus making the person something different. When other people see it or experience it they feel the person is no more the same.. or in other words a stranger.

I think joker means that what doesnt kill you, (like the thing he puts in the bankers mouth) makes you a different (strange) person than what you had been...
Possibly because those who have had a near death experience usually have a different (strange) perspective about life.

Expln. Build on: Dictionary meaning of strange i.e. unusual, uncommon or different from the normal course..

In my opinion, it was a very simple reference to the childhood wound caused to Joker by his father...His father deformed his face using a simple kitchen knife...of course, that does not kill him but made him...a stranger....

the childhood wound caused to Joker by his father He gave several explanations for his wounds, so we don't know whether or not this is true, I doubt it seeing as they didn't have any record of The Joker but social services would definitely spot a child whose face had been cut up like that. Furthermore, your answer is basically identical to pt18cher's answer from last month.
– Crow T RobotNov 25 '14 at 8:31

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